For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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1answer
63 views

Existence Proof: $T(v_i)=w_i$ for all $i=1,2,3,\dots,n$

Theorem to prove: Let $\{v_1,\dots,v_n\}$ be a linearly independent set in a finite-dimensional vector space $V$ and let $w_1,\dots,w_n$ be arbitrary vectors in a vector space $W$. Then there exists ...
1
vote
2answers
95 views

Revisited: $[T]^{\gamma}_{\beta}$ is diagonal?

Let $V$ and $W$ be finite-dimensional vector spaces with $\dim(V)=\dim(W)$ and let $T:V \rightarrow W$ be a linear map. How do I prove that there are bases $\beta$ of $V$ and $\gamma$ of $W$ such that ...
-1
votes
1answer
75 views

Matrix Polynomial Question

Suppose $A$ is a matrix with complex coefficients. Suppose $f(x)$ is a polynomial of minimal positive degree with property that $f(A)=0$. Let $P_A(x)$ be characteristic polynomial of $A$. Prove that ...
4
votes
1answer
171 views

Solving $Ax = b$ when $A$ is singular

I have a system of equations, expressed as $\mathbf{A} \begin{pmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} 0 \\ i (\frac{1}{2} + C - a) \\ i(\frac{1}{2} - C - a) \frac{m ...
2
votes
3answers
131 views

symmetric positive definite matrices

Why must a symmetric positive definite matrix must be invertible? I'm reading a proof of the Levi-Civita theorem in differential geometry but the author states this without proof and I haven't been ...
1
vote
0answers
78 views

Does this matrix have any properties?

The matrix is: $\left( \begin{array}{cc} \sin{\theta} & \cos{\theta} \\ \cos{\theta} & \sin{\theta} \\ \end{array} \right) $ I'm interested in its effect on points in the first quadrant, ...
1
vote
1answer
135 views

How to show by induction that, for $0<\theta<\pi$, $\det A_n=\frac{\sin (n+1)\theta}{\sin \theta}.$

I need help with the underlined part. Thanks in advance Let $A_n$ be the $n\times n$ matrix given by $$a_{ij}= \begin{cases} 0 & \text{if }|i-j|>1, \\ 1 & \text{if }|i-j|=1, ...
3
votes
2answers
3k views

Prove that the only eigenvalue of a nilpotent operator is 0?

I need to prove that if $\phi : V \rightarrow V$ is nilpotent, then its only eigenvalue is 0. I know how to prove that this for a nilpotent matrix, but I'm not sure in the case of an operator. How ...
3
votes
3answers
65 views

diagonalisability of matrix few properties

What are the most important things one should remember to check the diagonalisability of a matrix? Please help, I have exams on next week. Say some best and easy methods,time efficient.
0
votes
2answers
146 views

Linear Algebra : find the kernel of this transformation.

Q. I think I find the kernel but several... which is correct? Seems like depending on which variable I put as kernel, I can get several kernels. Correct? T is the transformation from $\mathbb{R}^2$ ...
0
votes
1answer
69 views

How to write this proof formally?

I have to prove that if $V$ is an unitary vector space and $W,U$ are subspaces of $V$, then $W^\bot \cap U^\bot=(W\land U)^\bot$ where $\bot$ means orthogonal complement and $\land$ is conjunction of ...
0
votes
1answer
79 views

Linear approximation of matrix norm

Given a square matrix $X=[x_1...x_N]$, and can be vectorized by $y=vec(X)=[x_1^T ... x_N^T]^T$ Is there any linear function can approximate $|| X ||$ (any matrix norm is okay) by using $y$?
0
votes
2answers
407 views

Transpose of matrix inverse: $(AA^T)^{-1}A^Tb \stackrel{?}{=} (A^TA)^{-1}A^Tb$

Given the matrix equation: $$ x^TA^TA = b^TA $$ I'm trying to find the least squares solution (i.e.; trying to minimize $r=||Ax-b||$). The matrix $A$ is not necessarily symmetric. When I solve it ...
0
votes
1answer
117 views

Derivative of matrix inner product

Let $A$ be a matrix with no restrictions, then I can compute, $$\nabla_x (x^t Ax) = \nabla_x (\sum_{i,j}x_i A_{ij} x_j) = Ax+A_Nx$$ Where $A_N$ is equal to $A$ on its diagonal entries and zero ...
0
votes
2answers
77 views

Linear Algebra: how do I know this is linear transformation?

T is the transformation from $\mathbb{R}^2$ to $\mathbb{R}^3$ $$T \left(\begin{array}{cc} x_{1}\\x_{2}\\\end{array}\right) = x_{1}\left(\begin{array}{cc} 1\\2\\3\end{array}\right) + ...
3
votes
2answers
67 views

Linear algebra, eigenvectors problem

Suppose you know that A is $2x2$ and symmetric. Assume the eigenvalues are $4$ and $7$. An eigenvector for $4$ is the vector $(3, -4)$. What is an eigenvector for $7$? So first we let ...
0
votes
1answer
328 views

Linear Transformation Reflection Equation

Linear Transformation $$T(\vec{x}) =\left(\begin{array}{cccc} 0.6&0.8\\0.8&-0.6\\ \end{array}\;\begin{array}{c}\end{array}\right)\vec{x}$$ is a reflection about a line L. I need to find the ...
1
vote
2answers
83 views

Matrix with same image and kernel

Does exists a matrix A for which kernel of A is the same as the image of A? Answer is True. But I couldn't find the example. I think I saw it from somewhere but I can't find it. It was 2 by 2 ...
1
vote
1answer
56 views

Linear Algebra True False: Inconsistency

Q. The system $A\vec{x} = \vec{b}$ is inconsistent if and only if $rref(A)$ contains a row of zeros. Answer is False. But my answer was True because I was thinking of the following example. ...
4
votes
1answer
106 views

$M_n(D)$ has only finitely many right ideals if and only if $n = 1$ or $D$ is finite.

Let $D$ be a division ring. Then prove that $R = M_n(D)$ has only finitely many right ideals if and only if $n = 1$ or $D$ is finite. I know that the ideals of $M_n(D)$ are of the form $M_n(I)$, ...
1
vote
1answer
72 views

Matrix and Linear Algebra True or False problem.

Q. True or False: If matrix A is a reduced row-echelon form, then at least one of the entries in each column must be 1. It comes down to this question. Can I have the following as Reduced ...
0
votes
2answers
115 views

Linear Equations using matrix and variables on the line

Q. I need a find a system of linear equations with three unknown variables whose solutions are the points on the line through (1,1,1) and (3,5,0). $ \frac{x-1}{2} = \frac{y-1}{4} = \frac{z-1}{-1}$ ...
2
votes
1answer
73 views

Linear Equation unknown variables and number of equations

Q. True of False The four linear equations with Three unknown variables is always inconsistent? Is it true or false? I thought of this example $$\left(\begin{array}{ccc} ...
5
votes
1answer
101 views

What is the isomorphism function in $M_m(M_n(\mathbb R))\cong M_{mn}(\mathbb R)$?

What is the isomorphism function in $M_m(M_n(\mathbb R))\cong M_{mn}(\mathbb R)$. I tried this $[[a_{ij}]_{kl}]\mapsto[a_{ijkl}]$ , but I couldn't prove all steps.
2
votes
2answers
62 views

A commutator problem

Let us consider $N \times N$ complex matrices, with $N >2$. Let D be a diagonal matrix, with $$D_{kk} = \sin \left(\frac{2\pi k}{N}\right), \space k = 0,..N - 1$$ I am looking for two ...
2
votes
1answer
49 views

how to prove $2^{n-1}|\det(A)$ where $A=[a_{ij}]\in M_n(\mathbb R)$ and $a_{ij}\in\{-1,1\} $

let $A=[a_{ij}]\in M_n(\mathbb R)$ such that $a_{ij}\in\{-1,1\} $ then how prove $$2^{n-1}|\ \det(A)$$ Thanks in advance
3
votes
1answer
94 views

Find the smallest square matrix in which some objects fit following some rules

I have to put some objects in a matrix. The data of these objects is given in another matrix in which each line contains an object, and the first column represents its width, and the second its ...
0
votes
1answer
868 views

Rank normal form of a matrix

There is a standard result in matrix theory that goes like this: Suppose $A$ is an $m\times n$ matrix of rank $r$, then there exist two non-singular matrices $E$ (of size $m\times m$) and $F$ (of size ...
2
votes
1answer
57 views

are normal subgroups of $SL(2,\mathbb{Z})$ also normal under the action of integer matrices in $GL(2, \mathbb{Q})$?

Ie, if $\Gamma\subseteq\text{SL}_2(\mathbb{Z})$ is a normal subgroup, and $\alpha\in\text{GL}_2(\mathbb{Q})\cap M_2(\mathbb{Z})$, then is $\alpha\Gamma = \Gamma\alpha$? (if necessary we can assume ...
-1
votes
1answer
116 views

by finding the eigenvalues and eigenvectors the evaluate the following.

so the question is : by finding the eigenvalues and eigenvectors of the matrix $$ P=\begin{bmatrix}1&6\\0&-2\end{bmatrix}\qquad\text{evaluate }P^{20}\begin{bmatrix}-2\\1\end{bmatrix} $$ I ...
3
votes
1answer
574 views

what does it mean for a matrix to be greater than another?

I am reading these notes on viscosity solutions, here is a theorem: Let us assume $u\in C^2$ is a classical solution of $F(x,u,Du,D^2u)=0$, $x\in \Omega$ then $u$ is a viscosity solution whenever ...
1
vote
1answer
218 views

How do I write this $t^2$ as a linear combination of polynomials in the basis?

I have this homework problem that says "In $\mathbb P_2$ find the change of coordinates matrix from the basis $\mathcal B=\{1-3t^2, 2+t-5t^2,1+2t\}$ to the standard basis. Then write $t^2$ as a linear ...
0
votes
1answer
62 views

$[T]^{\gamma}_{\beta}=[v]_{\gamma}$ with $\beta=\{1\}$ a basis for $F$

Let $V$ be a finite-dimensional vector space over $F$ with basis $\gamma$ and let $v\in V$. Find a linear map $T:F \rightarrow V$ such that $[T]^{\gamma}_{\beta}=[v]_{\gamma}$, where $\beta = \{1\}$ ...
1
vote
1answer
2k views

How Does One Find A Basis For The Orthogonal Complement of W given W?

I've been doing some work in Linear Algebra for my course at school. I just want to be clear about how to find the orthogonal complement of a subspace. The basis for the subspace, W, is shown below, ...
8
votes
1answer
84 views

*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$

I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form $A\in ...
0
votes
1answer
37 views

Equal Shape: Recovering an Isomorphism Between $M_{3\times 2}(F)$ and $P_5(F)$

I'm asked to find an isomorphism between $M_{3\times 2}(F)$ and $P_5(F)$, but what does it mean for a $3\times 2$ matrix to have an inverse?
1
vote
0answers
129 views

Extracting hitting times from the pseudoinverse of a Laplacian matrix for an undirected graph

Provided a pseudoinverted Laplacian matrix for an undirected graph $G$, how can I extract first passage and commute times between vertex pairs in $G$?
9
votes
6answers
2k views

If $A^2$ is invertible, then $A$ is also invertible?

True or False: If $A^2$ is invertible, then $A$ is also invertible. ($A$ is a matrix here.) The answer is true. I was trying to come up with an example that makes this false. But I couldn't. ...
1
vote
2answers
66 views

Number of Solutions in Linear System

$$\left(\begin{array}{cccc} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&0 \end{array}\;\middle\vert\;\begin{array}{c}2\\3\\4\\0\end{array}\right)$$ This is a $4$ ...
2
votes
1answer
91 views

elementary but confounding question about integer matrices (related to hecke operators)

Let $\Gamma(N)$ denote the kernel of the reduction map $\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ Let $p$ be a prime that is $1$ mod $N$, and let $M$ be the set of ...
2
votes
1answer
487 views

Commuting in Matrix Exponential

Let $A$, and $B$ be commuting $n\times n$ matrices, i.e. $A.B = B.A$. Let \begin{equation} \exp(A) = \sum_{i=0}^\infty\frac{1}{i!} A^i \end{equation} show that $\exp(A+B) = \exp(A).\exp(B)$.
1
vote
2answers
194 views

How to prove the existence of solution of a non linear system of equations

Writing the ortogonality condition for any element of O(n), I've arrived to: If we take n=2, we know that $\Lambda\Lambda^{T}=\mathbb{I}$, so: $$\begin{pmatrix} x & y \\ z & t \end{pmatrix} ...
1
vote
3answers
1k views

Is there any standard notation for specifying dimension of a matrix after the matrix symbol?

I want to explicitly specify dimension of matrices in some expressions, something like $$\boldsymbol{A}_{m \times n} \boldsymbol{B}_{n \times m} = \boldsymbol{C}_{m \times m} \, .$$ Is there any ...
0
votes
1answer
159 views

Augmented Matrix with a constant in 'A'

I have an augmented matrix defined: $$\left[\begin{array}{ccc|c} 1& 0& 2& 1\\ 0& 1& -1& 2\\ 1& -2& k+4& 5 \end{array}\right]$$ ...
1
vote
1answer
190 views

Question related to diagonally dominant matrix

A matrix is said to be positive if each entry in the matrix is positive. If $A$ is real, irreducible, diagonally dominant (or strictly dominant matrix) and has positive diagonal and non-positive ...
0
votes
1answer
53 views

Stability of a matrix

Suppose the hermitian part $H$ of a complex matrix $A$ be defined by $H=\frac{A+A^\ast}{2}$ and the skew hermitian part $S$ by $S=\frac{A-A^\ast}{2}$. If the hermitian part $H$ of $A$ is negative ...
1
vote
1answer
41 views

Combining elimination matrices

I am trying to combine several elimination steps into one matrix: more specifically I try to come up with a 3 by 3 matrix that first subtracts row 1 from row 2, subtract row 1 from row 3 and then ...
0
votes
2answers
164 views

Gram-Schmidt verifying orthonormal basis

Gram-Schmidt If I have an orthonormal basis, how do I verify that they are indeed orthonormal? I have Q, R and A is it enough to times Q` by Q to give me I? or A=QR? Edit: Let's say I ...
0
votes
1answer
388 views

Finding upper triangular matrix

I have this question, and im not sure I know how to solve it. "Find an upper triangular $U$ (not diagonal) with $U^2 = I$ which gives $U=U^{-1}$". Anybody who can help me getting the first steps of ...
0
votes
1answer
76 views

Invertibility of a monotone matrix.

I have a question regarding monotone matrix. How to prove that monotone matrix is invertible?